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given by x ↦ adx is a representation of a Lie algebra and is called the adjoint representation of the algebra. (Its image actually lies in Der. See below.)

Within End, the Lie bracket is, by definition, given by the commutator of the two operators:

where ○ denotes composition of linear maps.

If is finite-dimensional, then End is isomorphic to , the Lie algebra of the general linear group over the vector space and if a basis for it is chosen, the composition corresponds to matrix multiplication.

To be more precise, let G be a Lie group, and let Ψ: G → Aut(G) be the mapping g ↦ Ψg, with Ψg: G → G given by the inner automorphism

It is an example of a Lie group map. Define Adg to be the derivative of Ψg at the origin:

where d is the differential and TeG is the tangent space at the origin e (e being the identity element of the group G).

The Lie algebra of G is = TeG. Since Adg ∈ Aut, Ad: g ↦ Adg is a map from G to Aut(TeG) which will have a derivative from TeG to End(TeG) (the Lie algebra of Aut(V) being End(V)).

Then we have

The use of upper-case/lower-case notation is used extensively in the literature. Thus, for example, a vector x in the algebra generates a vector fieldX in the group G. Similarly, the adjoint map adxy = [x,y] of vectors in is homomorphic to the Lie derivativeLXY = [X,Y] of vector fields on the group G considered as a manifold.